Gravitational gradiometry is the measurement of the gravitational gradient field of differential accelerations between two infinitesimally close spatial points. The gravitational gradient field is described by a second rank tensor, Tij:Tij=−∂i,j2V(x,y,z)  (1)wherein i, j=(x, y, z) and the scalar V is the gravitational potential of a local reference frame of orthogonal Cartesian coordinates (x, y, z). Taking the z axis as pointing vertically into the ground, the components of the tensor at some point in the local reference frame (x, y, z), calculated by determining the spatial rate of change along directions x, y and z of the spatial rate of change of the gravitational potential in directions x, y and z, represent the rate of change of acceleration due to gravity along that direction. For example, the component Tyz represents the rate of change along the y direction of acceleration due to gravity along the direction z towards the ground, and is typically measured in units of Eötvös Units (1 Eötvös=1 EU=10−9 s−2). The tensor consists of nine components, only five of which are totally independent due to their geometrical symmetry (i.e. Tij=Tji, where i≠j) and due to the validity of the Laplace equation (i.e. Txx+Tyy+Tzz=0) for gravitational potential fields outside of the extent of gravitational field sources.
Providing apparatus that enables accurate and absolute measurements of the various components of the gravity gradient tensor Tij is very important in the fields of oil, gas and mining of various other natural resources. Gravitational gradiometry particularly enables the mapping of variations in the density of subsurface rocks and deposits to assist in the targeting of prospecting, and in increasing the effectiveness of drilling for oil and gas and mining. Gravitational gradiometry finds further application in defence and space industries for navigation and reconnaissance (e.g. void detection), geological prospecting, sub-sea/underwater navigation and exploration, terrestrial and marine archaeology, medicine and space exploration (for example, obtaining density maps of asteroids and other solar system orbital bodies).
For many gravity gradiometry applications, it is the Tzz component (i.e. the second order derivative of gravitational potential in the vertical direction) that many gradiometers aim to measure, whether by direct measurement, or by measuring at least some of the other tensor components and recalculating Tzz from their dependent relationship, or both. However, in their paper ‘On the combined gravity gradient modeling for applied geophysics’, Journal of Geophysics and Engineering, 2008, Vol 5, pp 348-356, Veryaskin and McRae show that by measuring and using the two off-diagonal gravity gradient tensor components Txz and Tyz, it is possible to obtain more information about anomalous subsurface density contrasts than by measuring and using the vertical gravity gradient component Tzz. To retrieve this subsurface density information, a gradiometer arrangement is required that is capable of simultaneously producing real-time data sets of direct measurements of both the Txz and Tyz tensor components.
A method of absolute measurement of gravity gradient tensor components was invented first by Baron Loránd von Eötvös as early as 1890, utilising a torsion balance with proof masses hung at different heights from a horizontal beam suspended by a fine filament. The gravity gradients give rise to differential forces being applied to the masses which result in a torque being exerted on the beam, and thus to angular deflection of the masses which can be detected with an appropriate sensor. A sensitivity of about 1 EU can be reached but measurement requires several hours at a single position due to the necessity to recalculate the gravity gradient components from at least five independent measurements of an angular deflection each with a different azimuth angle.
Practical devices, which have been built in accordance with this basic method of Eötvös, are large in size, bulky and have low environmental noise immunity, thus requiring specially prepared conditions for measurements. This excludes any possibility of using them on a moving carrier or for many practical applications where there are weight or space constraints, such as in the confined environment of a borehole, and in airborne drones, space launcher payloads, satellites, and extraterrestrial rovers.
Another method for absolute measurement of gravity gradient tensor components which enhances the above method was invented by Forward in the 1960s (see U.S. Pat. No. 3,722,284 (Weber et al) and U.S. Pat. No. 3,769,840 (Hansen)). The method comprises mounting both a dumbbell oscillator and a displacement sensor on a platform which is in uniform horizontal rotation with some frequency Ω about the axis of the torsional filament. The dumbbell then moves in forced oscillation with double the rotational frequency, whilst many of the error sources and noise sources are modulated at the rotation frequency or not modulated (particularly 1/f noise). The forced oscillation amplitude is at a maximum when the rotation frequency satisfies the resonance condition 2Ω/=ω0, where ω0 is the angular resonant frequency, and the oscillator quality factor Q tends to infinity. Unlike the non-rotating method, this method enables one to determine rapidly the quantities Tyy−Txx and Txy by separating the quadrature components of the response using synchronous detection with a reference signal of frequency 2 Ω.
The same principles can be directly used, as proposed by Bell (see U.S. Pat. No. 3,564,921), if one replaces the dumbbell oscillator with two or more single accelerometers properly oriented on such a moving platform. There are no new features of principle in this solution to compare with the previous one except that the outputs of the pairs of accelerometers require additional balancing.
Devices have been built according to this method, but they have met more problems than advantages, principally because of the need to maintain precisely uniform rotation and the small displacement measurement with respect to the rotating frame of reference. The devices have reached a maximum working accuracy of about a few tens of Eötvös for a one second measuring interval, and they are extremely sensitive to environmental vibrational noise due to their relatively low resonant frequencies. The technological problems arising in this case are so difficult to overcome that the existing developed designs of rotating gravity gradiometers show a measurement accuracy which is much lower than the limiting theoretical estimates.
In WO-A-96/10759 a method and apparatus for the measurement of two off-diagonal components of the gravity gradient tensor is described. According to this document, the transverse deflection of a stationary flexible string with fixed ends in its second fundamental mode of oscillation (the ‘S’ mode, as shown in FIG. 10b) is coupled to an off diagonal gravity gradient, whilst its deflection in its first fundamental mode of oscillation (the ‘C’ mode, see FIG. 10a) is coupled to an effective (i.e averaged with a weight function along the string's length) transverse gravitational acceleration. In other words, a string with fixed ends is bent into its ‘S’ mode by a gravity gradient only, provided that it does not experience any angular movements. Therefore, by measuring absolutely the mechanical displacement of such a string which corresponds to the ‘S’ mode it is possible to measure absolutely an off-diagonal component (i.e. Txz or Tyz, for a string aligned along the z axis) of the gravity gradient tensor. While this document teaches the use of a one-dimensional ‘string’, any generic element having a width and depth much smaller than its length, for example, a flat ribbon, is suitable.
In this design for a gradiometer having a current-carrying string, or ribbon, of length l aligned along the z axis and having a uniform mass distribution per unit length along its extent, the displacement, y(z,t), of the string from its undisturbed position (i.e. the straight line joining its fixed points at both ends), for example in the y-direction of the local coordinate frame as a function of the z-position of a unit element, and of time, t, can be described by the following force balancing equation for a vibrating string. (N.B. A similar equation and following analysis is applicable to the orthogonal direction transverse to the string and to any number of other directions).
                                          η            ⁢                                                  ⁢                                          ∂                2                                            ∂                                  t                  2                                                      ⁢                          y              ⁡                              (                                  z                  ,                  t                                )                                              +                      h            ⁢                          ∂                              ∂                t                                      ⁢                          y              ⁡                              (                                  z                  ,                  t                                )                                              -                      YA            ⁢                                                  ⁢                                          Δ                ⁢                                                                  ⁢                l                            l                        ⁢                                          ∂                2                                            ∂                                  z                  2                                                      ⁢                          y              ⁡                              (                                  z                  ,                  t                                )                                                    =                                            -              η                        ⁢                                                  ⁢                                          g                y                            ⁡                              (                                  0                  ,                  t                                )                                              -                      η            ⁢                                                  ⁢                                          T                yz                            ⁡                              (                                  0                  ,                  t                                )                                      ⁢            z                    +                                    I              ⁡                              (                t                )                                      ⁢                                          B                x                            ⁡                              (                                  0                  ,                  t                                )                                              -                                    I              ⁡                              (                t                )                                      ⁢                                          B                xz                            ⁡                              (                                  0                  ,                  t                                )                                      ⁢            z                    +                      thermal            ⁢                                                  ⁢            noise                                              (        3        )            
The components on the right hand side of the equation represent the forces acting on the string (including gravitational and magnetic forces) in they direction, and the components on the left hand side of the equation represent the restoring string forces in the y direction.
The equation has the boundary conditions corresponding to the fixed ends of the string, i.e. y(0,t)=y(l,t)=0. In this equation η denotes the string's mass per unit length, h is the friction coefficient per unit length, the parameters Y, A and Δl/l are the string's Young modulus, the area of its cross section and the string's strain respectively. The quantity gy(0,t) is the absolute value of the y-component of the gravitational acceleration and Tyz(0,t) the corresponding gravity gradient tensor component along the string, both taken at the centre of the local coordinate frame chosen (i.e. z=0). The quantity I(t) is the current flowing through the string. It is well known that a conductor carrying a current I(t) in a non-uniform magnetic vector field of flux density B(x, y, z) is subject to force F=I (t) {n×B(x, y, z)}, where n is the unit vector in the direction of current flow, in this case the z direction. The quantities Bx(0, t) and Bxz(0, t) therefore represent the absolute values of the x-component of the magnetic field and the corresponding magnetic gradient tensor component along the string, respectively, both taken at the centre of the local coordinate frame chosen.
Since the string is subject to Brownian fluctuations, the corresponding thermal noise driving source is included on the right side of equation (3).
Of the gravitational force components of the equation (3), −ηgy (0, t) represents the force in they direction on the unit element of the string due to the acceleration due to gravity, and −ηTyz(0,t)z represents the force in they direction on the unit element of the string due to the change along the z direction in the acceleration due to gravity.
Applying Fourier analysis to the complex shape of the string caused by its interaction with the gravitational and magnetic field, the function y(z,t), can be described, in the range z=0 to z=l, by an infinite sum of sinusoidal functions of period 2 l, with appropriate coefficients cy(n,t). Thus a solution of force balance vibration equation (3), which satisfies the boundary conditions shown above, can be represented by the following sum (4) wherein each term in n corresponds to one of the string's natural vibrational modes.
                              y          ⁡                      (                          z              ,              t                        )                          =                              ∑                          n              =              1                        infinity                    ⁢                                                    c                y                            ⁡                              (                                  n                  ,                  t                                )                                      ⁢                          sin              ⁡                              (                                                                            π                      ⁢                                                                                          ⁢                      n                                        l                                    ⁢                  z                                )                                                                        (        4        )            
By substituting equation (4) into equation (3) and by multiplying its left-hand and right-hand sides by sin(πn′z/l), and then by integrating both sides over z from 0 to l, one can obtain the following differential equation (4) for cy(n,t).
                                                                        ⅆ                2                                            ⅆ                                  t                  2                                                      ⁢                                          c                y                            ⁡                              (                                  n                  ,                  t                                )                                              +                                    2              τ                        ⁢                          ⅆ                              ⅆ                t                                      ⁢                                          c                y                            ⁡                              (                                  n                  ,                  t                                )                                              +                                    ω              n              2                        ⁢                                          c                y                            ⁡                              (                                  n                  ,                  t                                )                                                    =                                                            2                                  π                  ⁢                                                                          ⁢                  n                                            ⁡                              [                                                                            (                                              -                        1                                            )                                        n                                    -                  1                                ]                                      ⁡                          [                                                                    g                    y                                    ⁡                                      (                                          0                      ,                      t                                        )                                                  +                                                      1                    η                                    ⁢                                      I                    ⁡                                          (                      t                      )                                                        ⁢                                                            B                      x                                        ⁡                                          (                                              0                        ,                        t                                            )                                                                                  ]                                +                                                    (                                  -                  1                                )                            n                        ⁢                                                            2                  ⁢                  l                                                  π                  ⁢                                                                          ⁢                  n                                            ⁡                              [                                                                            T                      yz                                        ⁡                                          (                                              0                        ,                        t                                            )                                                        +                                                            1                      η                                        ⁢                                          I                      ⁡                                              (                        t                        )                                                              ⁢                                                                  B                        xz                                            ⁡                                              (                                                  0                          ,                          t                                                )                                                                                            ]                                              +                      thermal            ⁢                                                  ⁢            noise                                              (        5        )            where the quantities
                              ω          n                =                                            π              ⁢                                                          ⁢              n                        l                    ⁢                                                    Y                ρ                            ⁢                                                Δ                  ⁢                                                                          ⁢                  l                                l                                                                        (        6        )            represent the string's natural frequencies; and r and p are the relaxation time and the volume mass density of the string respectively.
When n takes an even value (i.e. for those terms cy(n,t) of the infinite sum in equation (4) corresponding to anti-symmetric vibrational modes of the string having a node at z=l/2, the midpoint of the string), the force component of equation (5) involving gy(0,t) and Bx(0, t) is equal to zero and the force component being a function of the gravitational gradient tensor component Tyz and magnetic field gradient tensor component Bxz(0, t) remains. Thus, for anti-symmetric modes of the string (i.e. n=even), cy is dependent only on Tyz and Bxz(0, t) (and thermal noise).
In practice this means that the amplitude, cy, of the anti-symmetric sinusoidal components of the deflection of the string in the y-direction, y(z,t), is dependent only on the magnitude of the gravity gradient tensor component Ty and the magnetic field gradient tensor component Bxz(0, t).
The string has an effective mechanical bandwidth of oscillation limiting its displacement response to oscillations below a few kHz (even for extremely stiff strings). The force on the string due to the magnetic field gradient is dependent on the current carried in the string. Therefore, by not pumping the string with any current at all or by pumping the string with an alternating current well outside the mechanical bandwidth of the string, the string will effectively not be sensitive to magnetic field gradients because oscillations at such frequencies are damped. In this way a string sensitive only to the gravity gradient tensor component Ty is provided.
The mid point of the string, z=l/2, is the position of a node in all anti-symmetric vibrational modes of the string. If sensors are positioned symmetrically in the longitudinal direction with respect to this point, it will be possible to identify displacements of the string corresponding to the string's natural anti-symmetric vibrational modes while discounting displacements corresponding to the string's symmetric vibrational modes.
It is particularly advantageous if displacement sensors are positioned at z=l/4 and z=3 l/4, positions corresponding to the antinodes of the first anti-symmetric vibrational mode of the string, n=2 (the ‘S’ mode). At these points the displacement of the string corresponding to the ‘S’ mode is at a maximum and thus the gradiometer sensing signal will also be at a maximum, giving optimum sensitivity.
In WO 96/10759, two rectangular type pick-up coils in the form of a Superconducting Quantum Interference Device (SQUID) are arranged to detect the transverse displacement in a superconducting Niobium string held under tension at its ends inside a superconducting casing; the whole apparatus being cooled to 4.2K or less in a cryogenic liquid helium vessel. Solenoids arranged symmetrically at either end of the string are driven by an alternating signal having frequency Ω to induce an AC supercurrent in the string also having frequency Ω. The superconducting casing excludes the external magnetic field from the casing such that no magnetic field forces act on the string and the displacement of the string from its straight line configuration is in response to the gravitational field only. The two coils of the SQUID device are positioned proximate to the string and are located at symmetrical longitudinal positions one on either side of the mid-point of the string and are arranged in a circuit as two arms of a superconducting magnetic flux transformer. The AC supercurrent carried by the string induces a current in each coil of the SQUID device proportional to the displacement of the string at that point from its undisturbed position. If the positions and responses of the two coils are arranged such that the two arms of the magnetic flux transformer are perfectly balanced either side of the mid-point of the string, the response is in ‘anti-phase’ such that the symmetrical modes of the string (i.e. n=odd, including the dominant ‘C’ mode) do not produce any signal current in the flux transformer. For the anti-symmetric modes, the displacement response of the string is dominated by the n=2‘S’ mode and all higher modes can be ignored (or factored in to error sources); then it follows that the output voltage of the SQUID is an AC signal having frequency Ω and an amplitude that is proportional to the displacement of the string in the first anti-symmetric ‘S’ mode only, and hence, to the off-diagonal gravitational gradient component (in the example given above, Tyz(0,t)). The amplitude of this SQUID output signal is obtained by synchronous detection of the signal using the alternating signal driving the solenoids as a reference. A force feedback circuit is also provided which takes as an input the voltage output of the SQUID and induces in the string a feedback current formed from this voltage output to increase the sensitivity of the device to the gravitational gradient component. For, a gradiometer of this design having typical practical parameters, the theoretical minimum gravity gradient detectable is calculated as being 0.02 EU. The string-based gravitational gradiometer device is less sensitive to vibrational noise than the earlier rotating gradiometer designs and lends itself to deployment on a mobile platform where measurements can be taken to retrieve high resolution data of local differences in gravity gradient. However, deployment is problematic in that the linear and angular accelerations of the mobile platform affect the deformation of the string and the output of the device.
In WO 03/27715 the string based gradiometer design is developed further by providing a gravity gradiometer in which the string is in the form of a uniform metal strip or ribbon and is constrained to its rest position at its mid-point, with, for example, a rigid knife-edge mounted to the casing and touching the ribbon but not exerting any force thereon. This knife-edge restricts any movement of the ribbon at that point and adds another boundary condition with the effect that deformation of the string into all symmetric modes (i.e. n=odd) is limited while deformation into all anti-symmetric modes (i.e. n=even), including the dominant ‘S’ mode, is permitted. Notably, deformation of the ribbon into the otherwise dominant first order symmetric ‘C’ mode is significantly limited. This use of a ribbon arrangement in place of a string is such that the ribbon is more constrained in its movement making the output of the device less dependent on linear accelerations exerted on the device and more manageable. This makes the device more suitable for operation on mobile platforms. The device operates in a liquid nitrogen cryogenic bath at 77K which reduces the effects of thermal noise and increases mechanical stability. In place of a SQUID device, two pick-up coils are provided positioned symmetrically about a mid-point of the ribbon and arranged as two arms of a resonant bridge circuit tuned to the frequency of an alternating carrier signal supplied to the ribbon as an alternating current. The frequency of the AC carrier current pumped to the ribbon is above the mechanical bandwidth of the tensioned ribbon such that the ribbon's displacement response due to interaction forces with the ambient magnetic field is damped and the detected signal is dependent on the gravitational field only. The two coils are located at positions directly adjacent the antinodes of the first anti-symmetric mode of the ribbon (i.e. at z=L/4 and z=3 L/4) which correspond to the maximum displacement and increases the sensitivity of the response. A voltage signal is induced in the bridge circuit having the same frequency as the carrier signal, and having an amplitude that is a measure of the average deflection of the ribbon over a region situated around the L/4 and 3 L/4 positions. By synchronously detecting the voltage amplitude of the induced signal with reference to the carrier signal, the amplitude of the local off diagonal gravity gradient component can be retrieved. The response of the ribbon is modulated with a square wave by indirectly changing its stiffness between a high value and a low value. This is achieved by using a square wave signal to switch a negative feedback circuit arranged to periodically produce in the ribbon a current signal proportional to the output of the bridge circuit but in anti-phase or quadrature therewith such that the ribbon becomes stiffened and is forced to its rest position. In the high stiffness or tensioned state, the response of the detector to the gravity gradients is low, and in low stiffness or relaxed state the response of the detector to the gravity gradients is high. This modulated output is retrieved using a lock-in amplifier. Three sets of four single axis gradiometer modules are provided in an ‘umbrella’ arrangement to remove the effect of angular accelerations on the output of the combined device, which is capable of providing absolute and direct measurement of all gravitational gradient tensor components.
In these string based gravitational gradiometers, the ability of the string to simultaneously deform into the ‘S’ mode and the other ‘parasitic’ symmetric modes introduces a significant noise source into the gradiometer device. The sensitivity of the gradiometer can be increased by the two detector coils being well balanced either side of the mid-point of the string to cancel out these unwanted parasitic modes. However, this balancing does not eliminate the effect that the deformation of the string in its symmetric modes has on the detectors and these unwanted parasitic modes make a significant contribution to the noise level above which the gradiometer signal, contributed by the deformation of the string in its anti-symmetrical (i.e. n=even) modes of oscillation, must be detected.
As described above, WO 03/027715 discloses one solution to this problem by providing a knife-edge at the mid-point of the string to add another boundary condition by constraining the string at its rest position there. This acts to restrict the string from deforming in its symmetric modes, most notably its ‘C’ mode. However, the string remains susceptible to deformation in the ‘W’ mode of oscillation (illustrated in FIG. 10c), which is the linear sum of all residual symmetric mode deflections. While deflection in this ‘W’ mode is of a smaller amplitude than that of the total deflection of an unconstrained string without a knife edge due to all symmetric modes, it remains a significant noise source that can limit the operational sensitivity of the apparatus to gravitational gradient signals.